March 11, 2024
Uniquely suited to provide detailed information on psychological and behavioral processes over time at the within-person level
54 publications indexed on PubMed in 2000, rising to 984 and 1203 publications in 2020 and 2021.
Empirical example I
Capturing (suicidal) crisis dynamics
Most popular statistical models for (emotion) time series:
Statistical concern: multimodality and skewness are highly prevalent: “less than half of emotion measurements (14%–40% across studies) exhibited a unimodal symmetric distribution” (Haslbeck et al., 2023).
Suggests the presence of multiple states, and the need for time series models that accommodates multimodality, such as hidden Markov models.
Current repeated measures and time series models aggregate across time points
However, we want to infer:
Analyzing data from multiple individuals:
fit HMM to the data of each individual separately
fit one and the same HMM model to the data of all individuals
explain some of the differences using covariates (e.g., R package depmixS4)
Does not allow (quantification of) natural variation (i.e., heterogeneity) between individuals
Does not allow for individual specific dynamics over time
Including a multilevel framework for the HMM, enables simultaneous estimation:
Multinomial logistic regression to estimate transition probabilities \(\gamma_{nij}\)
\(\gamma_{nij} = \frac{\text{exp}(\alpha_{nij})}{1 + \sum_{{j} = 2}^m \text{exp}(\alpha_{ni{j}})} \quad\) where \(\quad \alpha_{nij} = \bar{\alpha}_{ij} + \epsilon_{\left[\alpha\right]nij}\)
In addition, for the variable & state dependent emission Normal means:
\(\quad {\mu_{nki} = \bar{\mu}_{ki} + \epsilon_{\left[\mu\right]nki}}\)
Adopt a Bayesian approach
Pairing fine grained ESM data with multilevel HMM, we
Empirical example II
Nonverbal interpersonal interaction in patient-therapist dyads
39 patient-therapist dyads
15 min annotated video recordings of therapy session
mHMMbayes
R package for multilevel hidden Markov models using Bayesian estimation
Advantages:
Disadvantages:
39 patient-therapist dyads
15 min annotated video recordings of therapy session
Fitting a 3 state multilevel hidden Markov model on the nonverbal data:
library(mHMMbayes) ## specifying general model properties: # number of states m m <- 3 # number of dependent variables n_dep <- 6 # number of output categories for each dependent var q_emiss <- c(3, 2, 2, 3, 2, 2)
## specifying starting values
# Start values of the TPM
start_TM <- diag(.9, m)
start_TM[lower.tri(start_TM) | upper.tri(start_TM)] <- .05
# Start values for the emission distribution
start_EM <- list(matrix(c(0.70, 0.20, 0.10, # vocalizing therapist
0.05, 0.90, 0.05,
0.25, 0.70, 0.05), byrow = TRUE, nrow = m, ncol = q_emiss[1]),
matrix(c(0.15, 0.85, # looking therapist
0.35, 0.65,
0.30, 0.70), byrow = TRUE, nrow = m, ncol = q_emiss[2]),
matrix(c(0.80, 0.20, # leg therapist
0.80, 0.20,
0.80, 0.20), byrow = TRUE, nrow = m, ncol = q_emiss[3])
# continue emission probs here for the three other variables
)
# Run a model without covariate(s) and default priors:
out_3st <- mHMM(s_data = MHMM_nonverbal,
data_distr = "categorical",
gen = list(m = m, n_dep = n_dep, q_emiss = q_emiss),
start_val = c(list(start_TM), start_EMc),
mcmc = list(J = J, burn_in = burn_in))
out_3st
## Number of subjects: 39 ## ## 5000 iterations used in the MCMC algorithm with a burn in of 2000 ## Average Log likelihood over all subjects: -2193.739 ## Average AIC over all subjects: 4447.478 ## ## Number of states used: 3 ## ## Number of dependent variables used: 6 ## ## Type of dependent variable(s): categorical
summary(out_3st)
## State transition probability matrix ## (at the group level): ## ## To state 1 To state 2 To state 3 ## From state 1 0.942 0.019 0.040 ## From state 2 0.021 0.928 0.051 ## From state 3 0.071 0.103 0.826 ## ## ## Emission distribution ( categorical ) for each of the dependent variables ## (at the group level): ## ## $T_Vocalizing ## Category 1 Category 2 Category 3 ## State 1 0.886 0.012 0.102 ## State 2 0.018 0.976 0.006 ## State 3 0.143 0.824 0.033 ## ## $T_Looking ## Category 1 Category 2 ## State 1 0.037 0.963 ## State 2 0.307 0.693 ## State 3 0.226 0.774 ## ## $T_Leg ## Category 1 Category 2 ## State 1 0.950 0.050 ## State 2 0.947 0.054 ## State 3 0.839 0.161 ## ## $P_Vocalizing ## Category 1 Category 2 Category 3 ## State 1 0.009 0.985 0.006 ## State 2 0.884 0.005 0.111 ## State 3 0.664 0.157 0.178 ## ## $P_Looking ## Category 1 Category 2 ## State 1 0.333 0.667 ## State 2 0.045 0.955 ## State 3 0.316 0.684 ## ## $P_Leg ## Category 1 Category 2 ## State 1 0.658 0.342 ## State 2 0.998 0.002 ## State 3 0.410 0.590
emiss_pop <- obtain_emiss(out_3st) emiss_pop
## $T_Vocalizing ## Category 1 Category 2 Category 3 ## State 1 0.886 0.012 0.102 ## State 2 0.018 0.976 0.006 ## State 3 0.143 0.824 0.033 ## ## $T_Looking ## Category 1 Category 2 ## State 1 0.037 0.963 ## State 2 0.307 0.693 ## State 3 0.226 0.774 ## ## $T_Leg ## Category 1 Category 2 ## State 1 0.950 0.050 ## State 2 0.947 0.054 ## State 3 0.839 0.161 ## ## $P_Vocalizing ## Category 1 Category 2 Category 3 ## State 1 0.009 0.985 0.006 ## State 2 0.884 0.005 0.111 ## State 3 0.664 0.157 0.178 ## ## $P_Looking ## Category 1 Category 2 ## State 1 0.333 0.667 ## State 2 0.045 0.955 ## State 3 0.316 0.684 ## ## $P_Leg ## Category 1 Category 2 ## State 1 0.658 0.342 ## State 2 0.998 0.002 ## State 3 0.410 0.590
gamma_pop <- obtain_gamma(out_3st) gamma_pop
## To state 1 To state 2 To state 3 ## From state 1 0.942 0.019 0.040 ## From state 2 0.021 0.928 0.051 ## From state 3 0.071 0.103 0.826
Transition probability matrix Gamma (prior specification is optional):
prior_gamma( m, # number of states in the model gamma_mu0, # hyper-prior mean values on the Multinomial logit intercepts gamma_K0 = NULL, # number of hypothetical subjects gamma_mu0 is based on gamma_nu = NULL, # degrees of freedom inverse Wishart hyper-prior on covariance gamma_V = NULL, # hyper-prior variance-covariance matrix of Inverse Wishart n_xx_gamma = NULL # number of level 2 covariates predicting gamma ) prob_to_int( prob_matrix # complete transition probability matrix gamma )
Categorical emission distribution (prior specification is optional):
prior_emiss_cat( gen, # general model properties m, n_dep, and q_emiss emiss_mu0, # hyper-prior mean values on the Multinomial logit intercepts emiss_K0 = NULL, # number of hypothetical subjects emiss_mu0 is based on emiss_V = NULL, # degrees of freedom inverse Wishart hyper-prior on covariance emiss_nu = NULL, # hyper-prior variance-covariance matrix of Inverse Wishart n_xx_emiss = NULL # number of level 2 covariates predicting emission distribution(s) ) prob_to_int( prob_matrix # complete categorical emission probability matrix )
Continuous emission distribution (prior specification is MANDATORY):
prior_emiss_cont( gen, # general model properties m and n_dep emiss_mu0, # hyper-prior mean values of the Normal emission distribution emiss_K0, # number of hypothetical subjects emiss_mu0 is based on emiss_V, # degrees of freedom inverse Gamma hyper-prior on BETWEEN subject variance emiss_nu, # hyper-prior variance of Inverse Gamma on BETWEEN subject variance emiss_a0, # shape values of IG hyper-prior on emission standard deviation^2 emiss_b0, # scale values of IG hyper-prior on emission standard deviation^2 n_xx_emiss = NULL # number of level 2 covariates predicting emission distribution(s) )
Inverse Gamma hyper-prior on variance of Normal emission distribution:
